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September  2021, 14(9): 3305-3318. doi: 10.3934/dcdss.2021080

$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

1. 

School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China

2. 

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

3. 

School of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050016, China

* Corresponding author: Shenzhou Zheng

Received  March 2020 Revised  January 2021 Published  September 2021 Early access  June 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (No. 12001160), Natural Science Foundation of Hebei Province (No. A2019205218), Science Foundation of Hebei Normal University (No. L2019B02). The second author is supported by National Natural Science Foundation of China (No. 12071021)

We prove a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear elliptic equations $ F(x, u, Du, D^{2}u) = f(x) $ with oblique boundary condition in a bounded $ C^{2, \alpha} $-domain for every $ \alpha\in (0, 1) $. Here, the nonlinearities $ F $ is assumed to be asymptotically $ \delta $-regular to an operator $ G $ that is $ (\delta, R) $-vanishing with respect to $ x $. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear parabolic equations $ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $ with oblique boundary condition in a bounded $ C^{3} $-domain.

Citation: Junjie Zhang, Shenzhou Zheng, Chunyan Zuo. $ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3305-3318. doi: 10.3934/dcdss.2021080
References:
[1]

T. AlbericoC. CapozzoliR. Schiattarella and L. D'Onofrio, G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137.  doi: 10.3934/dcdss.2019009.  Google Scholar

[2]

S.-S. Byun and J. Han, $W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150.  doi: 10.1016/j.jde.2019.09.018.  Google Scholar

[3]

S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages. Google Scholar

[4]

S.-S. ByunM. Lee and D. K. Palagachev, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571.  doi: 10.1016/j.jde.2015.11.025.  Google Scholar

[5]

S.-S. ByunJ. Oh and L. Wang, Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308.  doi: 10.1093/imrn/rnu203.  Google Scholar

[6]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.  doi: 10.2307/1971480.  Google Scholar

[7]

L. A. Caffarelli and Q. Huang, Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17.  doi: 10.1215/S0012-7094-03-11811-6.  Google Scholar

[8]

M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303.  doi: 10.1017/S0308210500026378.  Google Scholar

[9]

J. ChoiH. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.  doi: 10.3934/dcds.2018097.  Google Scholar

[10]

H. Dong and N. V. Krylov, On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068.  doi: 10.1080/03605302.2012.756013.  Google Scholar

[11]

H. DongN. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69.  doi: 10.1090/S1061-0022-2012-01231-8.  Google Scholar

[12]

L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  doi: 10.1512/iumj.1993.42.42019.  Google Scholar

[13]

M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321.  doi: 10.1007/s10231-007-0045-2.  Google Scholar

[14]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.  Google Scholar

[15]

N. V. Krylov, Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798.  doi: 10.1080/03605302.2020.1805462.  Google Scholar

[16]

T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. doi: 10.1016/j.matpur.2012.02.004.  Google Scholar

[17]

D. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967.  doi: 10.1007/s00205-017-1209-x.  Google Scholar

[18]

S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp. doi: 10.1016/j.jmaa.2019.123749.  Google Scholar

[19]

P. Marcellini, Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031.  doi: 10.3934/dcdss.2020155.  Google Scholar

[20]

E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252.  doi: 10.1080/03605300600634999.  Google Scholar

[21]

C. Scheven and T. Schmidt, Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507.  doi: 10.2422/2036-2145.2009.3.04.  Google Scholar

[22]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  doi: 10.1002/cpa.3160450103.  Google Scholar

[23]

N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.  doi: 10.4171/ZAA/1377.  Google Scholar

[24]

J. Zhang and S. Zheng, Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008.  doi: 10.1002/mana.201600497.  Google Scholar

[25]

J. ZhangM. Cai and S. Zheng, Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235.  doi: 10.1016/j.na.2018.10.013.  Google Scholar

show all references

References:
[1]

T. AlbericoC. CapozzoliR. Schiattarella and L. D'Onofrio, G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137.  doi: 10.3934/dcdss.2019009.  Google Scholar

[2]

S.-S. Byun and J. Han, $W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150.  doi: 10.1016/j.jde.2019.09.018.  Google Scholar

[3]

S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages. Google Scholar

[4]

S.-S. ByunM. Lee and D. K. Palagachev, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571.  doi: 10.1016/j.jde.2015.11.025.  Google Scholar

[5]

S.-S. ByunJ. Oh and L. Wang, Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308.  doi: 10.1093/imrn/rnu203.  Google Scholar

[6]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213.  doi: 10.2307/1971480.  Google Scholar

[7]

L. A. Caffarelli and Q. Huang, Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17.  doi: 10.1215/S0012-7094-03-11811-6.  Google Scholar

[8]

M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303.  doi: 10.1017/S0308210500026378.  Google Scholar

[9]

J. ChoiH. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374.  doi: 10.3934/dcds.2018097.  Google Scholar

[10]

H. Dong and N. V. Krylov, On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068.  doi: 10.1080/03605302.2012.756013.  Google Scholar

[11]

H. DongN. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69.  doi: 10.1090/S1061-0022-2012-01231-8.  Google Scholar

[12]

L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423.  doi: 10.1512/iumj.1993.42.42019.  Google Scholar

[13]

M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321.  doi: 10.1007/s10231-007-0045-2.  Google Scholar

[14]

C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206.  doi: 10.1016/j.aim.2012.07.033.  Google Scholar

[15]

N. V. Krylov, Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798.  doi: 10.1080/03605302.2020.1805462.  Google Scholar

[16]

T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. doi: 10.1016/j.matpur.2012.02.004.  Google Scholar

[17]

D. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967.  doi: 10.1007/s00205-017-1209-x.  Google Scholar

[18]

S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp. doi: 10.1016/j.jmaa.2019.123749.  Google Scholar

[19]

P. Marcellini, Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031.  doi: 10.3934/dcdss.2020155.  Google Scholar

[20]

E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252.  doi: 10.1080/03605300600634999.  Google Scholar

[21]

C. Scheven and T. Schmidt, Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507.  doi: 10.2422/2036-2145.2009.3.04.  Google Scholar

[22]

L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76.  doi: 10.1002/cpa.3160450103.  Google Scholar

[23]

N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164.  doi: 10.4171/ZAA/1377.  Google Scholar

[24]

J. Zhang and S. Zheng, Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008.  doi: 10.1002/mana.201600497.  Google Scholar

[25]

J. ZhangM. Cai and S. Zheng, Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235.  doi: 10.1016/j.na.2018.10.013.  Google Scholar

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